Problem: The graphs of $y=3-x^2+x^3$ and $y=1+x^2+x^3$ intersect in multiple points.  Find the maximum difference between the $y$-coordinates of these intersection points.
Answer: The graphs intersect when the $y$-values at a particular $x$ are equal.  We can find this by solving \[3-x^2+x^3=1+x^2+x^3.\]This simplifies to  \[2(x^2-1)=0.\]This has two solutions, at $x=1$ and $x=-1$.  The $y$-coordinates for these points are  \[1+1^2+1^3=3\]and \[1+(-1)^2+(-1)^3=1.\]The difference between these values is $\boxed{2}$.